In his 1988 paper on 2+1 dimensional gravity, Edward Witten writes the $\mathfrak{iso}(2,1)$ commutation relations as $$[J_a,J_b]=\varepsilon_{abc}J^c,\quad [J_a,P_b]=\varepsilon_{abc}P^c,\quad[P_a,P_b]=0\tag{2.9}$$ where $J_a\equiv\frac{1}{2}\varepsilon_{abc}J^{bc}$ are the $\mathfrak{so}(2,1)$ Lorentz generators, $P_a$ are the translation generators and indices are raised and lowered with the $(-++)$ Minkowski metric $\eta_{ab}$ and its inverse. Then he "generalizes" the above commutation relations to $$[J_a,J_b]=\varepsilon_{abc}J^c,\quad [J_a,P_b]=\varepsilon_{abc}P^c,\quad[P_a,P_b]=-\Lambda\varepsilon_{abc}J^c\tag{2.23}$$ where $\Lambda$ denotes the cosmological constant. The resulting commutation relations are claimed to be those of $\mathfrak{so}(2,2)$ (in the case that $\Lambda<0$) or $\mathfrak{so}(3,1)$ (in the case that $\Lambda>0$). I am only interested in the former case.
My questions: What exactly is the logic behind this generalization? Why does Witten's trick of making the translation generators noncommutative, turn the commutation relations of $\mathfrak{iso}(2,1)$ into those of $\mathfrak{so}(2,2)$? What might have been his reasoning?
EDIT: I think I see it. The trick is engineered to simply reproduce the cosmological constant term in eq. (2.22) using the same methods as in paragraph 2.1. The Lie algebra (2.23) is readily shown to be isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$ and Witten must have been aware of the fact that the latter is isomorphic to $\mathfrak{so}(2,2)$. I was confused because I became acquainted with Witten's construction through a review paper which only ever mentioned (2.23) ...
